Class Note for PHYS 142 at UA
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Date Created: 02/06/15
LECTURE 25COMPLETING THE ASYMPTOTIC CALCULATION Lecture plan We Will complete the asymptotic analysis including solving the homework problem from Friday and invoking the guiding principle FlRST TRANSFORMATION op THE RlEMANNnHlLBERT PROBLEM We began with the following RiemannnHilbert problem characterizing the polynomials agm orthogonal with respect to eiNW 3 RiemannHilbert Problem 1 ma a 2 X 2 mama Az AsuN wtth the paupeattes Analyticity Az Ls auatutu for z e C 1R aua tahes auuttuuuus bau hdmy uatues Ax Aea as 2 tends in x wtth x ER and z 6 Cir z 6 C Jump Condition The bau hdmy uatues Me aauueatea by the aetattau 1 ErNV lt1 AM Aeltmgt 0 1 Normalization The mama Az u hmmaltzed at z be as follows 2 0 2 lim Az 11 ease 0 an We have been following he calculations of 1 and 2 and in particular 4 The rst transformation was as follows De ne 1 1 lt3 92 110s snows 110s anewmm which is taken to be analytic in c 700 1 Using 92 we de ne a new matrix Valued function the new unknown 32 as follows 4 132 e 3Aze N9 3 We then Veri ed that B satis es a new RiemannnHilbert problem RiemannHilbert Problem 2 ma a 2 X 2 mama 132 Baum uath the paupeataes Analyticity 132 u aaatutu tea 2 e CR aha tahes auutauuuus bauhdmy uatues 134a Ba as 2 tends in x wtth x ER and z 6 Cir z 6 C Jump Condition The bauhdmy uatues aae auuaeatea by the aetataua Medanawl 5Nltelttgteelttgtnvlttgtne 5 BM B40 0 weaves Normalization The mama 132 u uumahzea at z 00 as follows 6 lim Bz 11 Using the properties of the function g outlined in the past 2 lectures we veri ed that the jump matrix for B takes one of the following forms fuels 1 for 1 lt s lt 1 0 enGs I engltsgtzeltsgt plsl ne for lsl 3 1 0 1 Where the function G is given by 1 7 Cz Ziri wgxdx It is purely imaginary on the interval 71 We then exploited the following factorization e nGs 1 1 0 0 1 1 O 0 61135 enGs 1 1 0 enGS 1 V7V0V together with the amazing fact that these matrices can be extended oil the real axis where the oscillations turn into decay So we then de ned Dz as follows 0 For 2 outside the lens shaped region surrounding the interval 711 Dz o For 2 within the upper lens shaped region we set Dz BZ U 24 o For 2 within the lower lens shaped region we set Dz Bzv 21 24 22 25 23 The matrix D now solves a new RiemanniHilbert problem RiemannHilbert Problem 3 Find Dz satisfying the following three conditions Analyticity Dz is analytic for z E C E and takes continuous boundary values Dz Dz with x E 2 Jump Condition The boundary values are connected by the relation 8 Dz DzVDz z E E Normalization The matrix Dz is normalized at z 00 as follows 9 lim Dz l The jump matrix VD is de ned as follows 0 For 2 E 24 U 25 we have VDz VBz o Forz E 22 we have 10 VDP1 o Forz E 21 we have VDz vz o Forz E 23 we have VDz v And it is clear that the new unknown D is analytic of the more complicated union of contours shown above Moreover given the above considerations the jump matrices satisfy the following important property For any 6 gt 0 the jump matrix VDz is exponentially Close to ll for all values of 2 whose distance from 711 is greater than 6 Homework Problem lgnoring all contours except the interval 711 solve the following Riemanni Hilbert problem RiemannHilbert Problem 4 Find satisfying the following three conditions 1 analyticity The matrix D is analytic in C 711 2 2 Normalization ll O as 2 A 00 3 Boundary values and jump relation 11gt 1w D7ltzgt 31 5 gt 1 The solution First observe that you can diagonalize the jump matrix with a constant matrix transfor mation 017 1 711 12 1ogt Fw3F F lti 72 So diagonalize the jump matrix via 13 F lDlFgt rilbrk gt And now noting that both the jump and asymptotic behavior of F IDlFgt are diagonal try to express this quantity as a diagonal 7 az 0 14 1F 1D1F 7 gt lt gt 0 W It follows that az solves a scalar RHP which can be solved exactly 7 Z 7 114 15 az W BUILDING A GLOBAL APPROXIMATION TO Dz The intuition which we have developed indicates that 16 mg 132 Devil should be a new unknown which has no jump across 711 and has jumps that are exponentially near to l for 2 in the contour E but bounded away from i1 and so maybe this quantity satis es the guiding principle we have spoken about in Lecture 21 and see the discussion in the RHPSurvey lecture posted on the website But we are not quite home yet the jump matrices are not uniformly near to l Indeed the jump matrices for Dz may be described as follows for 2 near 1 there is a similar story for 2 near 71 o For I 6 11 6 we have the following description note that 1 denotes the analytic extension of 3 to 3 and evaluated by taking a boundary value on 11 6 72WiNf 13 5015 17gt VDltacgt3J e f E gt o For I E 17 61 we have 18 was 701 1 o For 2 E 21 we have 1 0 19 VD lt hifo 1mm 1 gt 3 a For 2 6 23 we have 1 0 20 VD lt 52mef 13mm 1 gt In this last formula we have extended the same quantity wt to the lower halfeplane and again called it 1113 Now observe that if we evaluate any of these formulae at 1 we nd a jump matrix which is just not near ll Moreover the behavior is rather delicate as the quantity 1113 5 has the following description for 5 near 1 15 iih5 5 7 1 2 for 5 6 a and 5 near 1 15 ih5 5 7 1 2 for 5 6 E and 5 near 1 where the function h5 is analytic in a neighborhood of 5 1 real for 5 6 R and h1 gt 0 01 Airy functions Airy functions Here is a transformation which removes the N dependence from the local behavior of the jump matrices 21 342 2pm gimme We will verify that 2 is analytic for z in a neighborhood of z 1 and maps a disc of xed size centered at z 1 to a neighborhood of 4 0 The neighborhood in the plane is very large roughly 0N23 If we consider the jump matrices VD 2 but brought over to the plane they have the following form C32 0 1 argco A 0 1 0 7 argtna 1 1 0 27 argC S 1 0 4n 9mm eggs2 1 7 mg Now the idea here is as follows can we construct a matrix valued function which has exactly these jumps in the 4 plane Ifso can we port it back over to the 2 plane and use it as a parametrip in a neighborhood of z 1 9 REFERENCES 1 P Delft T Kriecherhauer K TeR McLaughlin s Venakides and X Zhou Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theorf Comm Pure Appl Math 52 13354425 1999 2 P Delft T Kriecherhauer K TeR McLaughlin s Venakides and X Zhou Strong asymptotics of orthogonal polyno mials with respect to exponential weights Comm Pure Appl Math 52 14914552 1999 3 A Fokas A its and A v Kitaev Discrete Painlev equations and their appearance in quantum gravity Commun Math Phys 142 313314 1991 4 T Kriecherhauer and K T 7R McLaughlin Strong Asymptotics of Polynomials Orthogonal with Respect to Freud Weights Int Math Res Not No 6 pp 299333 1999
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